Calculus, Geometry, and Probability in n Dimensions:

The Broken Stick Problem in Higher Dimensions∗

IGL Project Report, Spring 2015

Alex Page, Yuliya Semibratova, Yi Xuan, Eva Rui Zhang,

M. Tip Phaovibul (Project Leader),

A.J. Hildebrand (Faculty Mentor)†

Illinois Geometry Lab

University of Illinois at Urbana-Champaign

May 19, 2015

1 The Broken Stick Problem

This project was motivated by the following problem, which first appeared more than 150 yearsago in an exam at Cambridge University, and which has since become one of the classic puzzles inmathematics; see, for example, Goodman [2].

The Broken Stick Problem. A stick is broken up at two points, chosen at random along itslength. Show that the probability that the pieces obtained form a triangle is 1/4.

0.24 0.43 0.33

The Broken Stick Problem gives rise to the “broken stick model”, an important probabilistic modelthat arises in areas ranging from biology to finance and which has been shown to be a good matchfor a variety of real-world data sets, including twin births reported in the Champaign-UrbanaNews-Gazette, and intervals between aircraft crashes of U.S. Carriers (see [1]).

2 The Broken Stick Problem with n Pieces

In a previous IGL project [4] we considered n-piece versions of the broken stick problem and provedthe following result.

∗An expanded version of this report is being prepared for possible publication; see [7].†ajh@illinois.edu.

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The n-Piece Broken Stick Problem (see [4]). Consider an n-piece broken stick obtained bychoosing n−1 points independently and uniformly along the length of a stick and breaking the stickinto n pieces at these points.

0.141 0.416 0.272 0.171

(i) The probability that there exist three of the n pieces that can form a triangle is

1 −n∏

k=2

k

Fk+2 − 1,

where Fn is the n-th Fibonacci number.

(ii) The probability that any triple of pieces chosen from the n pieces can form a triangle is

1(2n−2n

) .3 The Broken Stick Problem in Three Dimensions

In this project we extend the original Broken Stick Problem to three dimensions, with trianglesreplaced by tetrahedra, and six pieces instead of three, corresponding to the six edges of a tetrahe-dron. Our main result is as follows:

The 3D Broken Stick Problem (see [7]). A stick is broken up into six pieces by choosing fivepoints independently and randomly along its length and breaking up the stick at these points.

(i) The probability that one can form at least one tetrahedron from the six pieces is 6.5279%,with an estimated accuracy of ±0.001%.

(ii) The probability, P (k), that one can form exactly k pairwise incongruent tetrahedra is givenby the table below.

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k P (k)

0 93.4721%

1 1.2213%

2 1.5196%

3 0.4220%

4 0.6541%

5 0.2187%

6 1.0883%

7 0.1058%

k P (k)

8 0.1873%

9 0.0783%

10 0.1177%

11 0.0416%

12 0.1342%

13 0.0350%

14 0.0796%

15 0.0262%

k P (k)

16 0.0741%

17 0.0204%

18 0.0621%

19 0.0251%

20 0.0379%

21 0.0193%

22 0.0280%

23 0.0161%

k P (k)

24 0.1297%

25 0.0050%

26 0.0225%

27 0.0047%

28 0.0097%

29 0.0029%

30 0.1408%

Part (i) resolves a question that has been discussed at length in the online forums math.

stackexchange.com and math.overflow.net; part (ii) settles a question of Malkevitch and Mussa[6] on the geometry of the tetrahedron.

To obtain these results, we made use of known results from the geometry of a tetrahedron (see[8]) to design an efficient algorithm that simulates a 6-piece broken stick and tests whether the sixpieces can form a tetrahedron. We implemented this algorithm in C++ and ran the code on theIllinois Computing Cluster, www.campuscluster.illinois.edu. Our primary run involved 6.4·109

simulations on the cluster. We repeated this run several times, as a check of the accuracy of theresults; see [7] for details.

4 Extensions and Related Questions

In addition to the above main result, we considered several extensions of this result, and somerelated questions; for details and further results see [7].

The n-piece 3D Broken Stick Problem. As a natural extension of the 3D Broken StickProblem one can ask for the probability that, in an n-piece broken stick (where n ≥ 6), there exist6 pieces that form a tetrahedron. This is analogous to part (i) of the n-piece Broken Stick Problemdescribed above. Our results are given in the following table.

# of pieces Prob.

6 6.53%

7 21.55%

8 41.91%

9 62.36%

10 78.82%

# of pieces Prob.

11 89.93%

12 95.90%

13 98.60%

14 99.61%

15 99.91%

The 3D Broken Stick Problem under Sequential Breaking. Instead of the classical brokenstick model, where the 5 breaking points are chosen simultaneously along the length of the stick,another natural way to break a stick into six random pieces is to perform the breaking sequentiallyas follows. First, choose a single random point along the length of the stick, and break up the stickat this point to get two pieces. Then choose the largest of these two pieces, and break it up at a

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random point along its length, to get a total of three pieces. Repeat this process three more times,choosing at each stage the largest of the pieces to break up, until 6 pieces have been obtained.

We showed that, under this model, the probability that a tetrahedron can be formed is 26.93 . . .%,or about four times larger than the probability under the classical broken stick model. The rela-tively large value of this probability can be explained by the fact that choosing the largest pieceto break at each stage has the effect of creating a more uniform distribution of the lengths of thepieces, which in turn increases the chances that a tetrahedron can be formed.

The 3D Broken Stick Problem with Independent Random Lengths. Another naturalway to create six “random” lengths is to choose each length independently and uniformly from theunit interval [0, 1]. In this model, the probability of forming a tetrahedron increases even further,to 33.82 . . .%.

5 Distance Geometry

Underlying the 3D Broken Stick Problem is the question of when six given lengths form the edgelengths of a tetrahedron. This question is a special case of the so-called Distance Geometry Problem,which asks whether, given

(n2

)positive numbers, one can find n points in space whose pairwise

distances are the given numbers. The Distance Geometry Problem, in turn, lies at the core ofDistance Geometry, a fascinating, though little-known subfield of geometry, with important real-world applications to areas such as molecular biology and wireless sensor networks; see [5] or [3].

Viewed in this context, the Broken Stick Problem and its 3D analog can be regarded as randomversions of the Distance Geometry Problem: Instead of asking for an algebraic criterion for theexistence of an appropriate configuration (e.g., a triangle or a tetrahedron) with prescribed edgelengths, these problems ask for probabilities that such a configuration exists if the edge lengths aregenerated by the broken stick method, or some other random process.

This point of view suggests the following generalization of the Broken Stick Problem to higherdimensions.

The n-dimensional Broken Stick Problem. Let n be an integer ≥ 3. A stick is broken up at(n2

)− 1 points, chosen independently and randomly along its length. What is the probability that

there exist n points in an appropriate space Rm whose pairwise distances are the lengths of the(n2

)pieces obtained?

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References

[1] A. W. Ghent and B. P. Hanna, Application of the “Broken Stick” formula to the prediction ofrandom time intervals, American Midland Naturalist 79(2) (April 1968), 273–288.

[2] G. S. Goodman, The problem of the broken stick reconsidered, Math. Intelligencer 30 (2008),43–49.

[3] T. F. Havel, Distance geometry: Theory, Algorithms, and Chemical Applications, Encyclopediaof Computational Chemistry (2002), 1.

[4] L. Kong, L. Lkhamsuren, A. Turner, A. Uppal, A.J. Hildebrand, Random Points, BrokenSticks, and Triangles, IGL Project Report, April 2013; http://www.math.illinois.edu/

~hildebr/ugresearch/brokenstick-spring2013report.pdf

[5] L. Liberti, Euclidean Distance Geometry and Applications, SIAM Review, 56(1) (2014), 3-69.

[6] J. Malkevitch, D. Mussa, The transition from two dimensions to three dimensions- some ge-ometry of the tetrahedron, Consortium Number 105, (2013, Fall/Winter), 1-5.

[7] A. Page, Y. Semibratova, Y. Xuan, E. Zhang, M. Tip Phaovibul, A.J. Hildebrand, The BrokenStick Problem in Higher Dimensions: From a Classic Puzzle to Modern Distance Geometry,paper in preparation.

[8] K. Wirth, A. Dreiding, Edge lengths determining tetrahedrons, Elem. Math. 64 (2009), 160-170.

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